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Let $i_{\ast}, i^{\ast}: Sh(X) \to Sh(Y)$ be a geometric morphism of topos. In the derived category $D(X)$ of abelian sheaves on $X$, we can consider the internal derived Hom: $R\mathcal{Hom}_{D(X)}(F,...
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$\def\R{\mathscr{R}} \def\O{\mathcal{O}}$Let $X$ be a topological space and let $\R$ be a sheaf of unital non-commutative rings over $X$. When $\R$ is commutative, there is much literature on ...
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This answer nicely summarizes what the projective objects in the category of chain complexes $Ch(C)$ of an abelian category $C$ are. Namely, tfae: a chain complex is projective it is a split exact ...
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Over $\mathbb{Z}$, it is classical that $\operatorname{Ext}^1_{\mathbb{Z}}(\mathbb{Z}/n\mathbb{Z},\, \mathbb{Z}/m\mathbb{Z}) \;\cong\; \mathbb{Z}/\gcd(n,m).$ I would like to understand how this ...
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I'm currently reading through Weibel's "An Introduction to Homological Algebra" (page 30), and he offers the following definitions (paraphrased): A (homological) $\delta$-functor between ...
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I have a question on Lemma 1.7 of Beilinson's "Notes on absolute Hodge cohomology" at https://www.ams.org/books/conm/055.1/862628/conm055.1-862628.pdf. The purpose of the lemma is to compute ...
nkym's user avatar
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In the following, the derived categories I consider are unbounded. Let $F\colon \mathcal{A}\to \mathcal{B}$ and $G\colon \mathcal{B}\to \mathcal{A}$ be functors of Abelian categories such that $F$ is ...
Luvath's user avatar
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Let $M, N$ be finitely generated modules over a commutative Noetherian ring $R$. There are at least three ways to compute the Tor-modules $\operatorname{Tor}^R_i(M, N)$ all of which gives isomorphic ...
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In Peter Scholze's lecture notes on analytic geometry, Theorem 4.7 states that Banach spaces and Smith spaces are internal dual (in the category of condensed $\mathbb{R}$ vector spaces) to each other, ...
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I have the following question. Let $\{C_i\}_{i\in I}$ be an inverse system of complexes with members in some abelian category $\mathcal{A}$ where all small limits exist + some technical conditions. ...
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$\def\F{\mathscr{F}} \def\O{\mathscr{O}} \def\G{\mathscr{G}} \def\H{\mathscr{H}}$Let $X$ be a ringed space, and let $\F\in K(X):=K(\O_X\text{-Mod})$ be a complex of $\O_X$-modules. If $\F$ is K-flat, ...
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Suppose that $X$ is a smooth variety. Let $i_A: A \hookrightarrow X$ and $i_B: B \hookrightarrow X$ be closed subvarieties of $X$. In what situations can one understand/calculate the sheaf $L (i_B)^* ...
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Suppose $f: X \to Y$ is a projective morphism of algebraic varieties and $\mathcal F$ is a coherent sheaf on $X$. Can one easily detect that $R^i f_* \mathcal F=0$ for all $i \geq 0$ (that is $Rf_* \...
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It is shown in the IHES lectures on analytic stacks that the category of solid abelian groups has $\prod_{n \in \mathbb{N}} \mathbb{Z}$ as a compact projective generator. I may have just missed it but ...
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$\def\sI{\mathcal{I}} \def\sO{\mathcal{O}}$I would like to ask for a clarification on this question. I'm sorry if this ends up being a triviality, but after having thought about it for a while, I don'...
Elías Guisado Villalgordo's user avatar
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Let $G$ be a group, and let $\{M_i\}_{i\in I}$ be an inverse system of $G$-modules. Under which conditions does $H_n(G;\varprojlim M_i)\cong\varprojlim H_n(G;M_i)$ hold? It is acceptable for me to ...
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Let $(R,\mathfrak m)$ be a commutative Noetherian local ring. Let $S$ be a module finite commutative $R$-algebra. Then, $S$ is semilocal and $\mathfrak m S\subseteq J(S)$, where $J(S)$ is the ...
strat's user avatar
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This is a question that I have initially asked on Stack Exchange (Original Question), where unfortunately it has not found an answer. Any help is very appreciated. I am currently trying to understand ...
ClemensB's user avatar
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I will begin with some context; the question itself is highlighted below. This is all for some notes I am writing personally on homological algebra, amongst other things. To construct derived functors,...
Carl-Fredrik Lidgren's user avatar
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Let $T\colon \mathcal{A}\to \mathcal{B}$ be an additive functor between abelian categories and assume $\mathcal{B}$ has limits. We define the first satellite functor $S_1T\colon \mathcal{A}\to \...
Juan C. Cala's user avatar
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In this math overflow page, the poster gives a proof of the statement "an additive left exact functor $F$ preserves quasi-isomorphisms between $F$-acyclic objects." I'm having trouble ...
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Let $(R, \mathfrak m, k)$ be a commutative Noetherian local ring. Then, is it true that $\mathbf R\text{Hom}_R(k, -)\otimes_R^{\mathbf L} k$ commutes with arbitrary co-products?
uno's user avatar
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Let $R$ be a commutative Noetherian ring. Let $M$ be an $R$-module. It is well-known that $M$ is finitely generated if and only if the functor $M\otimes_R (-)$ preserves arbitrary products (for ...
uno's user avatar
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Let $R$ be a commutative Noetherian ring. Let $\operatorname{D}_{sg}(R)$ be the singularity category of $R$, i.e., the Verdier localization of $D_b(\text{mod } R)$ by the thick subcategory of perfect ...
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In several expositions of $\infty$-categories, I read that singular cohomology of a topological space with integral coefficients is a sheaf valued in $D(\mathbb{Z})$, if we consider Top and $D(\mathbb{...
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Let $A$ be an abelian category. Suppose $i:A\to A$ is a fully faithful functor from $A$ to itself. I wonder when the functor will be an equivalence. If $A$ is a "nice" category, I think $i$ ...
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Let $f: X \to Y$ be a proper morphism whose fibers have different dimensions, in particular $f$ is not flat. Let $L$ be a line bundle on $X$. What conditions would be sufficient to be able to conclude ...
Yellow Pig's user avatar
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$\def\C{\mathsf{C}} \def\D{\mathsf{D}} \def\hoc{\mathsf{HoC}} \def\hod{\mathsf{HoD}} \def\L{\mathbf{L}} \def\R{\mathbf{R}}$I'm a little confused by [R, Remark 6.4.14]. Before stating the remark, I'll ...
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Here is a discussion about an incorrect theorem of Roos, later corrected, some counterexamples and so on. Reading over this, I was a bit shocked because it contradicted something from Weibel's ...
FShrike's user avatar
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Let $R$ be a commutative Noetherian ring. For an $R$-module $M$, let $0\to E^0_R(M)\to E^1_R(M)\to \ldots $ denote the minimal injective resolution of $M$. I have two questions: (1) If $I$ is an ...
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Given a geometric morphism $$f:\mathscr{F}\to\mathscr{E}$$ where $\mathscr{F},\mathscr{E}$ are toposes, we know that $f^*$ does not preserve internal homs, i.e. $f^*[X,Y]\ncong[f^*X,f^*Y]$. We do have ...
Cameron's user avatar
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I'm learning about injective resolutions and derived functor sheaf cohomology, and it seems that every source on injective resolutions gives no examples. I feel like just one good example would make ...
A. Kriegman's user avatar
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The short version of my question is: Suppose $X$ is a (reasonably nice) topological space such that $X = \bigcup_{n \ge 1} X_n$ for an increasing sequence of (closed) subspaces $X_1 \subset X_2 \...
jessetvogel's user avatar
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Let $\mathcal C$ be a full subcategory (closed under isomorphism also) of an additive category $\mathcal A$. Then, $\text{add}(\mathcal C)$ is the full subcategory of $\mathcal A$ consisting of all ...
Alex's user avatar
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1 answer
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Let $f : X \to Y$ be a finitely presented proper morphism. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Do the functors $R^i f_* \mathcal{F}$ preserve any of the following properties: (1) ...
Ben C's user avatar
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Let $R$ be a complete local Cohen--Macaulay ring with dualizing module $\omega$. Let $M$ be a perfect complex over $R$. If the homology of $\mathbf R\text{Hom}_R(M,\omega)$ is concentrated in a ...
Snake Eyes's user avatar
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Let $(R,\mathfrak m)$ be a Gorenstein local ring of dimension $1$. Let $M$ be an $R$-module (not finitely generated) such that $M\neq \mathfrak m M$ and there exists a non-zero-divisor $x\in \mathfrak ...
Snake Eyes's user avatar
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1 answer
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Let $f:A \to B$ be a flat morphism of commutative $p$-adic completely rings. We denote by $D_{\text{comp}}(A)$ the derived category of complexes over $A$, which is derived $p$-adic complete. For a ...
OOOOOO's user avatar
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Assume that a functor on the category of groups vanishes on all projective objects. Is it necessarily the left derived functor of a half exact functor on this category?
Ali Taghavi's user avatar
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2 answers
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Let $k$ be a field of characteristic $0$. Let $R$ be a Noetherian local normal domain containing $k$. Also assume that $R$ is the homomorphic image of a Gorenstein ring of finite dimension, hence $R$ ...
Snake Eyes's user avatar
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Let $F:A\to B$ be an exact functor of Grothendieck abelian categories. If $F$ preserves injective objects, then does the exact functor $F:K(A)\to K(B)$ preserves K-injective complexes? For example, ...
Doug Liu's user avatar
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1 answer
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As far as I know, the inverse limit and its derived functors can be defined even in case we are dealing with a functor $F: I \to A$ from a category $I$ that is not cofiltered. I would content myself ...
Matteo Casarosa's user avatar
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In Mukai's paper Duality between $D(X)$ and $D(\hat{X})$ with its application to Picard sheaves, Nagoya Math Journal, 1981, there is one sentence that puzzles me. Let $X$ be an abelian variety over an ...
Doug Liu's user avatar
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Consider the following category $\mathbf{GrpMod}^*$ of compatible pairs, that is: an object is a pair $(G,M)$, where $G$ is a group and $M$ is a left $\Bbb Z[G]$-module. A morphism $(G,M) \to (H,N)$ ...
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Let $\mathcal A, \mathcal B,\mathcal C$ be abelian categories and let $F:\mathcal A \to \mathcal B,G: \mathcal B \to \mathcal C$ be left exact functors such that $RF:D(\mathcal A) \to D(\mathcal B), ...
Lukas Heger's user avatar
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Consider a scheme $X$ over athe complex numbers, $j:U\to X$ an open subscheme, $i:Z\to X$ its closed complement, and a perverse sheaf $F$ over $U$ with complex coefficients. The intermediate extension ...
W.Rether's user avatar
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6 votes
2 answers
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Let $(R,\mathfrak m)$ be a regular local ring. Let $I,J$ be proper ideals of $R$ such that $R/(I+J)$ has finite length i.e. $\sqrt{I+J}=\mathfrak m.$ Since $I+J$ annihilates $\text{Tor}_n^R(R/I, R/J)$ ...
Alex's user avatar
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1 answer
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Let $I$ be a directed set and let $X_I$ be a corresponding inverse system of, say, (complex) vector spaces or abelian groups (in my case in general not finite-dimensional, resp. not finitely generated)...
AlexE's user avatar
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Let me recall the following definition. Let $F: C \to D$ be a functor between homotopical categories. Denote by $\gamma_C: C \to \mathrm{Ho} C$ the localization and similary for $D$. A total left ...
carciofo21's user avatar
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Mike Shulman, in his article Homotopy limits and colimits and enriched homotopy theory observes (towards the end of page 8) that it may be the case that a functor $F: C \to D$ between homotopical ...
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